Data Transformation in Two‐way Analysis of Variance

ABSTRACT One purpose of data transformation is to better satisfy the fundamental assumptions of statistical analysis by linear models: (a) additivity, (b) homogeneity of variance, and (c) normality. If the original data do not satisfy (a)‐(c), a nonlinear transformation may improve approximation to these ideal conditions. This paper considers estimation of the parameter λ of the family of power transformations for data conforming to a replicated two‐way crossed classification. The transformation family y (λ) defined by though having the advantage of being continuous through λ = 0, is shown to yield likelihood procedures which are scale invariant only if the design matrix allows for removal of an additive constant. Several alternative definitions for (ii) are given which yield both scale invariant likelihood procedures and continuity through λ = 0. By using (ii) in the case of a linear regression through the origin, numerical examples show how point estimates of λ can be shifted and how confidence intervals on λ can be shrunk and stretched almost arbitrarily, simply by rescaling the original observations. Two procedures for choosing a transformation for data from a replicated two‐way crossed classification are compared by empirical sampling. The statistic Lmax(λ), proposed by Box and Cox (1964), is based upon the likelihood of the transformed observations. The statistic Z(λ), proposed in this paper, is a linear combination of test statistics for removable nonadditivity and variance trending with mean. The null distribution of Z(λ) is approximately N(0,l). Two cases of empirical sampling from a 3 × 4 crossed classification with four replications were studied. In samples from the truncated Gaussian distribution, the coefficient of variation was roughly 0.23. In samples from the truncated Cauchy distribution, the coefficient of variation was roughly 0.91. The points of truncation depended upon the cell population means. The scale for the Cauchy distribution with location 0 was chosen so that the 95% points matched those of N(0,l). In Gaussian Samples: The 5% test based on Lmax(λ) was extremely conservative, the actual size being more like 0.3%. The estimated size of the nominal 5% test based on Z(λ) was 4.4% In Cauchy Samples: The 5% test based on Lmax(λ) was very non‐conservative. The estimated size was 15%. The estimated size of the nominal 5% test based on Z(λ) was 5.6%. A procedure for choosing a weighting for the components of Z(λ) based upon the sample data is also derived and investigated. Sampling shows that it works well for the two cases studied. This procedure should also work well in practice.